Quantum Model Theory (QMod): Modeling Contextual Emergent Entangled Interfering Entities

  • Diederik Aerts
  • Sandro Sozzo
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7620)

Abstract

In this paper we present Quantum Model Theory (QMod), a theory we developed to model entities that entail the typical quantum effects of contextuality, superposition, interference, entanglement and emergence. The aim of QMod is to put forward a theoretical framework that is more general than standard quantum mechanics, in the sense that, for its complex version it only uses this quantum calculus locally, i.e. for each context corresponding to a measurement, and for its real version it does not need the property of ‘linearity of the set of states’ to model the quantum effect. In this sense, QMod is a generalization of quantum mechanics, similar to how the general relativity manifold mathematical formalism is a generalization of special relativity. We prove by means of a representation theorem that QMod can be used for any entity entailing the typical quantum effects mentioned above. Some examples of application of QMod in concept theory and macroscopic physics are also considered.

Keywords

Quantum modeling contextuality interference QMod 

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References

  1. 1.
    Aerts, D.: Example of a macroscopical situation that violates Bell inequalities. Lett. N. Cim. 34, 107–111 (1982)CrossRefGoogle Scholar
  2. 2.
    Aerts, D.: A Possible Explanation for the Probabilities of Quantum Mechanics. J. Math. Phys. 27, 202–210 (1986)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Aerts, D., Durt, T., Grib, A., Van Bogaert, B., Zapatrin, A.: Quantum structures in macroscopical reality. Int. J. Theor. Phys. 32, 489–498 (1993)MATHCrossRefGoogle Scholar
  4. 4.
    Aerts, D.: Quantum Structure in Cognition. J. Math. Psych. 53, 314–348 (2009)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Aerts, D.: Quantum Particles as Conceptual Entities: A Possible Explanatory Framework for Quantum Theory. Found. Sci. 14, 361–411 (2009)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Aerts, D.: Interpreting Quantum Particles as Conceptual Entities. Int. J. Theor. Phys. 49, 2950–2970 (2010)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Aerts, D.: A Potentiality and Conceptuality Interpretation of Quantum Physics. Philosophica 83, 15–52 (2010)Google Scholar
  8. 8.
    Aerts, D., Gabora, L.: A Theory of Concepts and Their Combinations I&II. Kybernetes 34, 167–191, 192–221 (2005)MATHCrossRefGoogle Scholar
  9. 9.
    Aerts, D.: Being and Change: Foundations of a Realistic Operational Formalism. In: Aerts, D., Czachor, M., Durt, T. (eds.) Probing the Structure of Quantum Mechanics: Nonlinearity, Nonlocality, Probability and Axiomatics, pp. 71–110. World Scientific, Singapore (2002)CrossRefGoogle Scholar
  10. 10.
    Gabora, L., Aerts, D.: Contextualizing Concepts Using a Mathematical Generalization of the Quantum Formalism. J. Exp. Theor. Art. Int. 14, 327–358 (2002)MATHCrossRefGoogle Scholar
  11. 11.
    Nelson, D.L.: Entangled Associative Structures and Context. In: Bruza, P., Lawless, W., van Rijsbergen, K., Sofge, D. (eds.) Proceedings of the Association for the Advancement of Artificial Intelligence (AAAI) Spring Symposium 8: Quantum Interaction, March 26-28. Stanford University, Stanford (2007)Google Scholar
  12. 12.
    Gabora, L., Rosch, E., Aerts, D.: Toward an Ecological Theory of Concepts. Ecol. Psych. 20, 84–116 (2008)CrossRefGoogle Scholar
  13. 13.
    Flender, C., Kitto, K., Bruza, P.: Beyond Ontology in Information Systems. In: Bruza, P., Sofge, D., Lawless, W., van Rijsbergen, K., Klusch, M. (eds.) QI 2009. LNCS, vol. 5494, pp. 276–288. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  14. 14.
    Gabora, L., Aerts, D.: A Model of the Emergence and Evolution of Integrated Worldviews. J. Math. Psych. 53, 434–451 (2009)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    D’Hooghe, B.: The SCOP-formalism: An Operational Approach to Quantum Mechanics. In: AIP Conference Proceedings, vol. 1232, pp. 33–44 (2010)Google Scholar
  16. 16.
    Aerts, D., Czachor, M., Sozzo, S.: A Contextual Quantum-based Formalism for Population Dynamics. In: Proceedings of the AAAI Fall Symposium (FS-10-08), Quantum Informatics for Cognitive, Social, and Semantic Processes, pp. 22–25 (2010)Google Scholar
  17. 17.
    Veloz, T., Gabora, L., Eyjolfson, M., Aerts, D.: Toward a Formal Model of the Shifting Relationship between Concepts and Contexts during Associative Thought. In: Song, D., Melucci, M., Frommholz, I., Zhang, P., Wang, L., Arafat, S. (eds.) QI 2011. LNCS, vol. 7052, pp. 25–34. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  18. 18.
    Aerts, D., Sozzo, S.: Quantum Structure in Cognition: Why and How Concepts Are Entangled. In: Song, D., Melucci, M., Frommholz, I., Zhang, P., Wang, L., Arafat, S. (eds.) QI 2011. LNCS, vol. 7052, pp. 116–127. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  19. 19.
    Young, T.: On the Theory of Light and Colours. Phil. Trans. Roy. Soc. 92, 12–48 (1802); Reprinted in part in: Crew, H. (ed.): The Wave Theory of Light, New York (1990)Google Scholar
  20. 20.
    de Broglie, L.: Ondes et Quanta. Comptes Rendus 177, 507–510 (1923)Google Scholar
  21. 21.
    Schrödinger, E.: Quantizierung als Eigenwertproblem(Erste Mitteilung). Ann. Phys. 79, 361–376 (1926)MATHCrossRefGoogle Scholar
  22. 22.
    de Broglie, L.: La Nouvelle Dynamique des Quanta. In: Proceedings of the Solvay Conference-1928, Electrons et Photons, pp. 105–132 (1928)Google Scholar
  23. 23.
    Jönsson, C.: Elektronen Interferenzen an Mehreren Künstlich Hergestellten Feinspalten. Zeit. Phys. 161, 454–474 (1961)CrossRefGoogle Scholar
  24. 24.
    Feynman, R.P.: The Feynman Lectures on Physics. Addison-Wesley, New York (1965)MATHGoogle Scholar
  25. 25.
    Jönsson, C.: Electron Diffraction at Multiple Slits. Am. J. Phys. 4, 4–11 (1974)CrossRefGoogle Scholar
  26. 26.
    Arndt, M., Nairz, O., Vos-Andreae, J., Keller, C., van der Zouw, G., Zeilinger, A.: Wave-particle Duality of C 60 Molecules. Nature 401, 680–682 (1999)CrossRefGoogle Scholar
  27. 27.
    Aerts, D.: Quantum Structures due to Fluctuations of the Measurement Situations. Int. J. Theor. Phys. 32, 2207–2220 (1993)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Aerts, D.: Quantum Structures, Separated Physical Entities and Probability. Found. Phys. 24, 1227–1259 (1994)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Aerts, D., Aerts, S.: The Hidden Measurement Formalism: Quantum Mechanics as a Consequence of Fluctuations on the Measurement. In: Ferrero, M., van der Merwe, A. (eds.) New Developments on Fundamental Problems in Quantum Physics, pp. 1–6. Springer, Dordrecht (1997)CrossRefGoogle Scholar
  30. 30.
    Aerts, D., Aerts, S., Coecke, B., D’Hooghe, B., Durt, T., Valckenborgh, F.: A Model with Varying Fluctuations in the Measurement Context. In: Ferrero, M., van der Merwe, A. (eds.) New Developments on Fundamental Problems in Quantum Physics, pp. 7–9. Springer, Dordrecht (1997)CrossRefGoogle Scholar
  31. 31.
    Aerts, S.: Hidden Measurements from Contextual Axiomatics. In: Aerts, D., Czachor, M., Durt, T. (eds.) Probing the Structure of Quantum Mechanics: Nonlinearity, Nonlocality, Probability and Axiomatics, pp. 149–164. World Scientific, Singapore (2002)CrossRefGoogle Scholar
  32. 32.
    Aerts, S.: The Born Rule from a Consistency Requirement on Hidden Measurements in Complex Hilbert Space. Int. J. Theor. Phys. 44, 999–1009 (2005)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Aerts, D., Sozzo, S.: Entanglement of Conceptual Entities in Quantum Model Theory (QMod). In: Busemeyer, J.R., Dubois, F., Lambert-Mogiliansky, A. (eds.) QI 2012. LNCS, vol. 7620, pp. 114–125. Springer, Heidelberg (2012)Google Scholar
  34. 34.
    Aerts, D., Gabora, L., Sozzo, S.: How Concepts Combine: A Quantum Theoretic Modeling of Human Though. Accepted for publication in Topics in Cognitive Science (2012)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Diederik Aerts
    • 1
  • Sandro Sozzo
    • 1
  1. 1.Center Leo Apostel (CLEA)Vrije Universiteit Brussel (VUB)BrusselsBelgium

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